Physical
Manifestations
of
Periodic
Functions
Matthew Koss
College of
the Holy Cross
July 12, 2012
IQR Workshop: Foundational Mathematics
Concepts for the High School to College Transition
Simple Block and Spring
Data Studio 500
Simple Harmonic Motion
Simple Harmonic Oscillations
x  t   A cos (w t f )
x  t   A sin (w t  f )
or
or
y  t   A cos (w t f )
y  t   A sin (w t  f )
A
wt+f
f
w
T
f
Amplitude
Phase (radians)/Angle (radians)
Phase Constant (radians)
Angular Frequency (rad/s)
Period (s)
Frequency (Hz)
Simple Harmonic

Motion
y (t )  A cos 
for Block and
Spring
1.5

k
t  f 
 m

Y (meters)
1
f 
T
w  2 f
1
0.5
0
0
0.5
1
1.5
2
2.5
-0.5
-1
w
k
m
-1.5
y (t )  AX Postition
cos (w(meters)
t f )
Another Representation
 2

x(t )  A cos 
t f 
 T

or
 2

y (t )  A cos 
t f 
 T

A
Amplitude
2
t f
T
Total Angle ( )
f
Initial Angle
T
Period
or
x(t )  A cos  2 ft f 
or
y (t )  A cos  2 ft f 
A
2 ft  f
f
f
Amplitude
Total Angle ( )
Initial Angle
Frequency
Review
A Periodic Function (sine or cosine) is the Recorded History of
the Oscillations of an object attached to a spring.
xmax 
xmin 
  2
t  T
Position, velocity, and acceleration
If you know calculus
 2

y (t )  A cos 
t f 
 T

d
d
 2

v(t )  x(t )  A cos 
t f  
dt
dt
 T

d
a (t )  v(t ) 
dt
 2

y  A cos 
t f 
 T

Calculus Approach
dy d
 2

v
 A cos 
t f  
dt dt
 T

2 A  2
 2
 2

 A sin 
t f 

sin 
t f 
T
 T
 T
 T

d 2 y dv d 2
 2

a 2 

A sin 
t f  
dt
dt dt T
 T

2
 2
 2
 2 
 2

A cos 
t f 
 
t f 
 A cos 
T
 T
 T
 T 
 T

2
If Not, then …
 2

x(t )  A cos 
t f 
 T

2
 2

v(t )  
A sin 
t f 
T
 T

1
f 
T
w  2 f
k
m
w

k
 2 
 2
  2 
a (t )   
t f  

 A cos 

m
 T 
 T
  T 
2
2
Zero Offset
• Oscillations do not always occur about the zero point.
• To account for this, there is one additional term called the
zero offset which is middle value in the oscillations.
• So, more completely:
y (t )  A cos (w t f )  yoffset
or
x(t )  A cos (w t f )  xoffset
iPads
and
Video Physics
Physics Toolkit
Atom Can Execute Simple Periodic Motions
States of Matter Simulation
SHM is the Projection of Circular Motion
Illustration
y(t)
A
A
y(t)
y2(t)
y1(t)
y 2 y1
PhET Rotation Simulation
Simple Pendulum
FT
T  2
L
g
mg
 (t )  A cos(wt  f ), w 
g
L
PhET Pendulum Simulation
Physical Pendulum
Same as a simple pendulum, but…
L  Distance from pivot to cm or cg.
mgL
w
I
I
T  2
mgL

L

cm
axis
Oscillations on a String
y (t )  A cos  2 ft f 
y ( x, t )  A( x) cos  2 ft f 

 n
y ( x, t )   A sin 
 L


x   cos  2 ft f 

Tangent on Traveling Waves
A wave is a disturbance in position propagating in time.
Many traveling waves are periodic in both position and time, e.g.
A
v

2
 2

y  A sin 
x
t f 
T
 

Mathematical Relationships
In general: y  f ( x, t )
and
y  f ( x  vt )
Specifically:
Periodic
Sine Waves
A
kxwt+f
w
T
f
k

2
 2

y  A sin 
x
t f 
T
 

y  A sin(kx  wt  f )
Amplitude
Phase (radians)
Angular Frequency (rad/s)
Period (s)
Frequency (Hz)
(Angular) Wave number
Wavelength
v  wave speed v 
or v   f , v  w / k

T
T  period
1
f 
w  2 f
T
  wavelength
k
2

Waves and Oscillations Compared
y  x, t   A sin(kx  wt  f )
y  t   A sin (w t  f )
An oscillation in time is a “history” of a wave at a particular place.
An oscillation in space is a “snapshot” of a wave at a particular time,
y  x, t   A sin(kx  wt  f )
y  t   A sin(kxspecific  wt  f )
 A sin(wt   ),   kxspecific  f
y  x   A sin(kx  wtspecific  f )
 A sin(kx   ),   wtspecific  f
Sum of Two Traveling Waves Makes
Standing Waves
Last Slide
of
Digression
Standing Waves on a
String, or
Oscillations on a String
y(t )  A( x)cos  2 ft f 
n
fn 
2L
FT
L
f  f1
, n  1, 2,3,
f  2 f1  f 2
1
f1 
2L
FT
L
f n  nf1 , n  1, 2,3,
f  3 f1  f3
String Vibrates the Air
Guitar Strings
The strings on a guitar can be
effectively shortened by
fingering, raising the
fundamental pitch.
The pitch of a string of a
given length can also be
altered by using a string of
different density.
Sound is a Periodic Oscillation of the Air
t 0
v

B
v



2
v
T
t
2
Tuning Forks
Data Studio 500 Redux
Beats
If the two interfering oscillations have different frequencies they will
superimpose, but the resulting oscillation is more complex. This is
still a superposition effect. Under these conditions, the resultant
oscillation is referred to as a beat.
amplitude (m)
2
1
0
-1
0
50
100
am plitude (m )
-2
2
150
200
250
150
200
250
Time (sec)
1
0
-1
0
50
100
-2
Time (sec)
amplitude (m)
2
1
0
-1
0
50
100
150
200
250
150
200
250
150
200
250
-22
am plitude (m )
Time (sec)
1
0
-1
0
50
100
-2
2
amplitude (m)
Time (sec)
1
0
-1
0
50
100
-2
Time (sec)
Beat Frequency Mathematics
fBeat = f1 -f2
I1 (t )  I sin(2 f1t ) & I 2 (t )  I sin(2 f 2t )
I sin(2 f1t )  I sin(2 f 2t )
  2 f1t    2 f 2t  
  2 f1t    2 f 2t  
 2sin 
cos



2
2




 2  f 2  f1  
 2 ( f1  f 2 ) 
I beat (t )  2 I sin 
t  cos 
t
2
2




amplitude (m)
2
1
0
-1
0
50
100
150
-2
Time (sec)
200
Amplitude (I) of Sound Oscillations
The loudness of a sound is
much more closely related to
the logarithm of the intensity.
Sound level is measured in
decibels (dB) and is defined as:
I0 is taken to be the threshold
of hearing:
MacScope II
Audacity
iPads & I Phones
More Complex Sounds
Fundamental/Normal Modes
Time and Frequency Domains
Sample
Musical
Instrument
Sounds
in the
Frequency
Domain
Web References/Resources
PhET Simulations
http://phet.colorado.edu/en/simulations/category/new
Springs
http://phet.colorado.edu/en/simulation/mass-spring-lab
Rotation
http://phet.colorado.edu/en/simulation/rotation
Atomic Oscillation
http://phet.colorado.edu/en/simulation/states-of-matter
Pendulum
http://phet.colorado.edu/en/simulation/pendulum-lab
Normal Modes
http://phet.colorado.edu/en/simulation/normal-modes
Making Waves
http://phet.colorado.edu/en/simulation/fourier
Video Physics
http://itunes.apple.com/us/app/vernier-video-physics/id389784247?mt=8
Physics Toolkit
http://physicstoolkit.com/
MacScope & Physics2000
http://www.physics2000.com/Pages/Downloads.html
Audacity
http://audacity.sourceforge.net/download/